# Proving that $∂U = \overline U \cap \overline{X\setminus U}$

I want to prove that $$∂U = \overline U \cap \overline{X\setminus U}.$$

Not too sure where to start with this question so any help would be appreciated.

I am also struggling to prove $U^{\circ} \subset U$ for any $U$. I have the following definitions: $$∂U = \overline U \setminus U^{\circ},\ U^{\circ} = X\setminus \overline{(X\setminus U)}.$$

Any help is much appreciated.

• Just to confirm $\ U^{\circ} = X\setminus \overline{(X\setminus U)}$ is the given definition of interior, right? – ir7 Mar 1 '18 at 15:20
• @ir7 yes sorry that is the definition of interior – Nicole Mar 1 '18 at 16:36

$$∂U = \overline U \setminus U^{\circ} =\overline U \setminus( X\setminus \overline{(X\setminus U)}) = \overline U \cap \overline{(X\setminus U)}.$$
$$U^{\circ} =X\setminus \overline{(X\setminus U)} \subset X\setminus (X\setminus U) = U,$$ as $X\setminus U \subset \overline{(X\setminus U)}$.